2 M-1.T 
2, * (1//M) ( 1 Ay Ag .… Ae )^; ku- 1,2;....M 
Ae z exp[2(k-n)ri/M]; n = integer, k= 1,2,...,M 
A and e,, k =1,2,...,M are the eigenvalues and eigenvectors of the 
unit displacement operator U. They are arbitrary to the extent that n 
is any integer. Two values of n are of special interest: n=0 results 
in that f and à are the usual M-dimensional discrete Fourier 
transforms of f and g; nz(M*1)/2, M odd, gives a real interpolation 
formula, i.e. if the components of g(0) are real, so are those of g(u) 
for arbitrary real yu. We will choose this latter value n=(M+1)/2 in 
order that the scalar products (11) be real for real y. 
As D is diagonal, (11) can be written 
M 
(8/8u) E ro exp[2(k-n)uri/M] = 0 (13) 
k=1 
This is an equation of the form 
M k * 
f Aat. 2 0; a, = (k-n)f,g (14) 
ek , k Kk 
k=1 
which, once it has been solved for E, gives 
zo 
um T log E (15) 
We now have two procedures for determining the matching parameter u: 
1) According to (6), we can compute the sum Lf(x)g(x+yu) for different 
values of y and simply choose that value y,, which gives a maximum for 
this sum. This method is the one generally used. 
2) According to (15), we can compute p from a knowledge of E. In order 
to determine E, we must solve a nonlinear equation having M roots. A 
great number of these will generally be real, each root corresponding 
to a local extremum of the scalar product £f.g(u). The necessary 
computational effort to solve the equation and choose the root 
corresponding to the translation between the images 1s generally 
greater than that needed for the procedure given in point 1). Also the 
second procedure involves an approximation which, as will be shown 
later, introduces unacceptable errors due to wrap-around. If no 
arguments besides those given above are to be considered, the 
procedure given in point 2) therefore appears to be of no advantage. 
EFFECTS OF FINITE IMAGE SIZE 
As the images are of finite size, the leftmost part of the left image 
will not appear in the right image. In the same way, the rightmost 
part of the right image will not appear in the left image. When the 
images are translated relative each other in order to match, the 
missing parts must be introduced in one way or another. The definition 
(10) of the unit displacement operator implies that the missing parts 
are introduced cyclically. This definition is of course not realistic 
and introduces wrap-around errors. In this section we investigate if 
these errors are serious, and possible methods of eliminating them. 
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